Geography Reference
In-Depth Information
Since in general N
2
>f
2
(7.56) indicates that inertia-gravity wave frequencies
must lie in the range f
N . The frequency approaches N as the trajectory
slope approaches the vertical, and approaches f as the trajectory slope approaches
the horizontal. For typical midlatitude tropospheric conditions,
inertia
-gravity
wave periods are in the approximate range of 12 min to 15 h. Rotational effects
become important, however, only when the second term on the right in (7.56) is
similar in magnitude to the first term. This requires that tan
2
α
≤|
ν
|≤
N
2
/f
2
10
4
,
∼
=
in which case it is clear from (7.56) that ν
N . Thus, only low-frequency gravity
waves are modified significantly by the rotation of the earth, and these have very
small parcel trajectory slopes.
The heuristic parcel derivation can again be verified by using the linearized
dynamical equations. In this case, however, it is necessary to include rotation. The
small parcel trajectory slopes of the relatively long period waves that are altered
significantly by rotation imply that the horizontal scales are much greater than
the vertical scales for these waves. Therefore, we may assume that the motions
are in hydrostatic balance. If in addition we assume a motionless basic state, the
linearized equations (7.37)-(7.40) are replaced by the set
∂u
∂t
−
∂p
∂x
=
1
ρ
0
fv
+
0
(7.57)
∂v
∂t
+
∂p
∂y
=
1
ρ
0
fu
+
0
(7.58)
∂p
∂z
−
θ
θ
1
ρ
0
g
=
0
(7.59)
∂u
∂x
+
∂v
∂y
+
∂w
∂z
=
0
(7.60)
∂θ
∂t
+
dθ
dz
=
w
0
(7.61)
The hydrostatic relationship in (7.59) may be used to eliminate θ
in (7.61) to yield
1
ρ
0
∂p
∂z
∂
∂t
N
2
w
=
+
0
(7.62)
Letting
u
,v
,w
,p
ρ
0
=
Re
ˆ
p
exp i (kx
νt)
u,
v,
ˆ
w,
ˆ
ˆ
+
ly
+
mz
−
and substituting into (7.57), (7.58), and (7.62), we obtain
ν
2
f
2
−
1
u
ˆ
=
−
(νk
+
ilf )
p
ˆ
(7.63)
